To solve this problem, we need to compute the derivative of the given function:

$f(x) = \sqrt{\frac{(x+3)^5}{x^3 e^{x-2}}}$

Step 1: Rewriting the function

We can rewrite the square root as a power of $\frac{1}{2}$:

$f(x) = \left( \frac{(x+3)^5}{x^3 e^{x-2}} \right)^{\frac{1}{2}}$

Step 2: Applying the Chain Rule and Quotient Rule

Let’s differentiate the function using the chain rule and quotient rule. First, define:

$g(x) = \frac{(x+3)^5}{x^3 e^{x-2}}$

We differentiate the outer part first, treating $g(x)^{1/2}$ as a composition of functions. The derivative of $g(x)^{1/2}$ is:

$\frac{d}{dx} g(x)^{1/2} = \frac{1}{2} g(x)^{-\frac{1}{2}} \cdot g'(x)$

Step 3: Derivative of $g(x)$

Now, we need to differentiate $g(x) = \frac{(x+3)^5}{x^3 e^{x-2}}$ using the quotient rule:

$g'(x) = \frac{d}{dx} \left( \frac{(x+3)^5}{x^3 e^{x-2}} \right)$

Using the quotient rule:

$g'(x) = \frac{\left[ 5(x+3)^4 \cdot e^{x-2} \cdot x^3 - (x+3)^5 \cdot (3x^2 e^{x-2} + x^3 e^{x-2}) \right]}{(x^3 e^{x-2})^2}$

Step 4: Simplification

After simplifying, the expression matches with the form presented in option b:

$f'(x) = \left( \frac{(x+3)^5}{x^3 e^{x-2}} \right)^{1/2} \times \frac{5}{2} \times \frac{1}{x+3}$

Thus, the correct answer is option d.

Would you like further clarification on any part of the derivative process?

Here are five related questions you could explore:

1. How does the chain rule work when applied to composite functions?
2. Can the quotient rule be used on any rational function?
3. What are the steps to simplify complex derivatives like this?
4. How do exponential terms affect the derivative process?
5. What is the importance of simplifying expressions before differentiating?

Tip: When solving complex derivative problems, break them into smaller parts and apply rules like the chain rule or quotient rule step by step.